Sep 17, 2018 - Euler-Lagrange equations of Lovelock actions in arbitrary dimensions. Here we study ... We focus on the Bach equation in dimension n = ...

0 downloads 5 Views 312KB Size

FORMAL POWER SERIES FOR ASYMPTOTICALLY HYPERBOLIC BACH-FLAT METRICS AGHIL ALAEE AND ERIC WOOLGAR Abstract. It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant Q-curvature. This allows us to determine the free data at conformal infinity, and to select those choices that lead to Einstein metrics. Interestingly, the mass is part of that free data, in contrast to the pure Einstein case. We then choose a convenient generalization of the Bach tensor to (bulk) dimensions n > 4 and consider the higher dimensional problem. We find that the free data for the expansions split into loworder and high-order pairs. The former pair consists of the metric on the conformal boundary and its first radial derivative, while the latter pair consists of the radial derivatives of order n − 2 and n − 1. Higher dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case. This is reflected in the relative complexity of the higher dimensional problem, but we are able to obtain a relatively complete result if conformal infinity is not scalar flat.

1. Introduction In seminal work, Fefferman and Graham [12, 13] studied formal series solutions of the Einstein equation for asymptotically hyperbolic metrics expanded about conformal infinity. This led to the identification of data for the singular boundary value problem for these metrics, the discovery of obstructions to power series solutions, and ultimately the discovery of new conformal invariants for the conformal boundary. It also laid the groundwork for holography within the AdS/CFT correspondence. More recently, Gover and Waldron [14] and Graham [15] have performed similar analyses for a scalar geometric PDE problem, a singular boundary value problem for the Yamabe equation. Albin [1] has announced an analysis of asymptotically hyperbolic formal series solutions of the Euler-Lagrange equations of Lovelock actions in arbitrary dimensions. Here we study the question of formal series expansions for a fourth-order geometric PDE in the asymptotically hyperbolic setting. We focus on the Bach equation in dimension n = 4, and on a slightly modified equation amenable to our methods when n > 4. The Bach equation is (1.1)

0 = Bac :=

1 1 ∇b ∇d Wabcd + Wabcd Rbd , (n − 3) (n − 2)

where Wabcd is the Weyl tensor, Rab is the Ricci tensor, and Bab is called the Bach tensor. On closed 4-manifolds, (1.1) is the Euler-Lagrange equation for the functional Z |Wg |2 dVg , (1.2) W(g) = M 1

2

AGHIL ALAEE AND ERIC WOOLGAR

though for n ≥ 5, critical points of this functional are all locally conformally flat and therefore satisfy (1.1) somewhat trivially. There are inequivalent ways to extend the Bach tensor, originally defined only for n = 4, to higher dimensions. The choice (1.1) can be motivated by noting that it is an integrability condition for conformally Einstein metrics (see, e.g., [8]). We seek solutions of (1.1) with a pole at infinity of order 2, expressible as 1 (1.3) g = 2 dx2 ⊕ hx , x on a complete n-manifold (M, g), or at least on a collar neighbourhood x < ǫ, where hx extends differentiably to x = 0 and induces a Riemannian metric on each constant-x hypersurface. Metrics obeying (1.3) are called conformally compactifiable and asymptotically hyperbolic. For small x, an open region in (M, dx2 + hx ) may be isometrically embedded as an open, bounded region in a product manifold in which the locus x = 0 becomes a boundary. This locus, equipped with the conformal class [h0 ], is called conformal infinity (see Section 2 for more terminology). When such a metric is Einstein, it is called Poincar´e-Einstein. We will use the term Poincar´eBach for Bach-flat metrics of the form (1.3). As with the Poincar´e-Einstein case [12, 13], we will pursue here the relatively modest goal of finding formal power series for hx for Poincar´e-Bach metrics. We do not consider convergence, not even on a collar of x = 0. Define Eg := Rcg +(n − 1)g , (1.4) Ag := trg Eg = Scalg +n(n − 1) , (we sometimes omit the subscript g) and recall that a conformally compactifiable and asymptotically hyperbolic metric has Eg = O(x). If Eg = O(x2 ), a calculation shows that (1.5)

h′x (0) = 0 ,

where we denote differentiation with respect to x by a prime. We recall (following terminology in [11]) that a conformally compactifiable metric is asymptotically hyperbolic Einstein to order k if Eg ∈ O(xk ) for x any special defining function; this is also called asymptotically Poincar´eEinstein (APE) to order k. Any metric that is APE to order 2k < n − 1 is partially even to order 2k, by which we mean that the odd-order derivatives h(2j−1) (0) vanish for j ≤ k. 1.1. Four bulk dimensions. A major motivation for the present paper is the assertion of Maldacena [18] that in n = 4 bulk dimensions one can replace Einstein gravity by classical conformal gravity, by which is meant the variational theory of the action functional (1.2) with suitable asymptotically anti-de Sitter or asymptotically hyperbolic fall-off conditions and other conditions. Maldacena’s proposal is that the condition h′x (0) = 0, together with certain physical considerations, selects only those critical points of this action which are Einstein. For another approach, based on Anderson’s formula [3] for renormalized volume but ultimately invoking other considerations as well, see [2]. It seems to us more satisfactory (and obviously more in the spirit of holography) instead to search for well-defined asymptotic conditions which alone can select Einstein metrics, at least when considering Riemannian signature metrics. This brings us to our first main result. Theorem 1.1. Let h0 be a Riemannian metric on Σ3 and let Φ, Ψ be smooth symmetric h0 tracefree (0, 2)-tensors on Σ such that divh0 Ψ = 0. Let Ti denote smooth functions on Σ for i ≥ 2. For any such data h0 , Φ, Ψ, Ti with i ≥ 2, the equations Bg = 0 admit a unique normal form solution (1.1) on (M 4 , g) with hx ≡ h(x) given by a formal power series in x,

3

such that (Σ, [h0 ]) = ∂∞ M is conformal infinity, with h(0) = h0 , h′ (0) = 0, tf h0 h′′ (0) = Φ, tf h0 h′′′ (0) = Ψ, and trh0 h(i) (0) = Ti . 1.2. Choosing the conformal representative and the mass aspect. In view of [18], one can try to find the subset of formal power series for Bach-flat metrics which are formal power series for Poincar´e-Einstein 4-metrics. Such metrics have h′ (0) = 0 and h′′ (0) = −2Ph0 , where Ph0 denotes the Schouten tensor of h0 . However, the 4-dimensional Bach tensor is conformally invariant. Its vanishing is an integrability condition for conformally Einstein metrics. To choose Einstein representatives within conformal classes of metrics, one must impose a further condition that will fix the trace data in Theorem 1.1. Now, Einstein metrics obviously have constant scalar curvature Scalg = −12 and constant Branson Q-curvature Qg = 6 where 1 (1.6) Qg := −∆g Scalg + Scal2g −3| Rcg |2g . 6 One can impose one of these conditions (constant Ag or constant Qg ) in order to fix the infinitely many trace data Ti (except, it turns out, T4 ) in Theorem 1.1, leaving finitely many data to be chosen by imposing conditions at infinity. To see that the condition A = 0 fixes a unique representative metric g within its conformal class of Bach-flat metrics, consider that if g˜ := u2 g and g both have scalar curvature −n(n − 1), 4

4 ∆g u + u (n−2) − 1 u = 0 then u must be a positive solution of the Yamabe equation − n(n−2) and u → 1 at conformal infinity. But then u ≡ 1 by the maximum principle. (We assume here completeness with no “inner” boundary—if one is present, there may sometimes be other solutions for u.) Since the condition Q = 6 fixes the same free data, it also fixes a unique representative metric g within its conformal class of Bach-flat metrics. It turns out that neither fixing Ag (and thus Scalg ) nor fixing Qg will determine T4 . Consider the quantity [20, 10, 21] 1 ′′ 2 1 ′′ 2 1 1 (4) (1.7) µ := trh0 h (0) − h (0) = T4 − h (0) . 3! 2! 3! 2! h0 h0

When conformal infinity carries a round sphere metric, this quantity is called the mass aspect function. In that case, if g is Poincar´e-Einstein the mass (the integral of µ over conformal infinity) must vanish [4], and indeed so must the mass aspect (e.g., [21, see the proof of Conjecture 2.7]). More generally, to select Poincar´e-Einstein metrics, we must choose the correct conformal class, and this was not completely achieved by choosing data as in Theorem 1.1. We must in addition impose the condition T4 = 32 |h′′ (0)|2h0 so that µ = 0.1

Corollary 1.2. Let Ψ be a symmetric (0, 2)-tensor on conformal infinity with trh0 Ψ = 0, divh0 Ψ = 0. A formal power series for an asymptotically hyperbolic 4-metricin normal form with h(0) = h0 , h′ (0) = 0, h′′ (0) = −2Ph0 , h′′′ (0) = Ψ, and 3!1 trh0 h(4) (0) = |Ph0 |2h0 is a formal solution of the system Bg = 0, Ag = 0 if and only if it is a formal solution of the Einstein equations. Corollary 1.3. Let Ψ be a symmetric (0, 2)-tensor on conformal infinity with trh0 Ψ = 0, divh0 Ψ = 0. A formal power series for an asymptotically hyperbolic 4-metricin normal form with h(0) = h0 , h′ (0) = 0, h′′ (0) = −2Ph0 , h′′′ (0) = Ψ, and 3!1 trh0 h(4) (0) = |Ph0 |2h0 is a

1There is debate over whether complete metrics can have vanishing mass but nontrivial mass aspect when

n = 4 and A ≥ 0 (see [9] for further details).

4

AGHIL ALAEE AND ERIC WOOLGAR

formal solution of the system Bg = 0, Qg = 6 if and only if it is a formal solution of the Einstein equations. 1 trh0 h(4) (0) = |Ph0 |2h0 (i.e., µ = 0) but one does fix all the other If one does not fix 3! data as in Corollary 1.2 or 1.3, one obtains for each choice of mass aspect function µ a formal power series for an asymptotically Poincar´e-Bach metric. Such a series can represent a metric of arbitrary mass. 1.3. Higher bulk dimensions. The Bach tensor is most naturally defined in 4-dimensions, where it has vanishing divergence and trace, is a local conformal invariant, and obstructs conformally Einstein metrics, while nontrivially generalizing the Einstein condition (i.e., there are Bach-flat metrics that are not Einstein). There are many inequivalent generalizations of the Bach tensor for n > 4 [8], each preserving some desirable properties of the 4-dimensional Bach tensor but none preserving them all. Despite this, the n > 4 case provides an opportunity to gain insight into higher-order geometric equations. We will observe an interesting “splitting” of the free data, which may be common in higher order geometric equations, as well as a delicate mechanism for fixing the conformal gauge which may be specific to our particular choice of Bach tensor generalization. The Bach tensor as defined by (1.1) is not conformally invariant for n ≥ 5 (see [8, equation 4.16]). More importantly for present purposes, the divergence of the tensor defined by (1.1) is not identically zero when n ≥ 5. Instead, it is given by (n − 4) 1 b (1.8) ∇ Bab = (gbc ∇a A − gac ∇b A) E bc . ∇a Ebc − ∇b Eac − (n − 2)2 2(n − 1) The quantity in square brackets is the Cotton tensor, written in terms of Eab . The vanishing of the right-hand side of (1.8) is a necessary integrability condition for solutions of B = 0. This imposes constraints on the otherwise-free data. This may be an advantage in other contexts, but it complicates the power series analysis and diverts attention from some of our main points. To apply the Fefferman-Graham procedure in the same manner as when n = 4, it is useful to preserve the vanishing of the divergence of the (generalized) Bach tensor. To this end, we note that by simple manipulations the right-hand side of (1.8) can be written as a divergence of a ˆ := B − X is divergence-free, symmetric (0, 2)-tensor Xab which vanishes when n = 4. Then B as desired (it is not in general tracefree, however), and reduces to B when n = 4. Proposition 1.4. Define (1.9)

ˆab := Bab − (n − 4) B 2(n − 2)2

n (n + 2) 2 c 2 AEab − 2Eac Eb + |E|g − A gab . (n − 1) 4(n − 1)

ˆab = 0. Then ∇b B Proof. Take the divergence of (1.9) and use (1.8).

ˆg = 0 yield Amongst higher dimensional generalizations of the Bach equations, the equations B to the Fefferman-Graham technique with minimal fuss while capturing key features common to many other suitable generalizations, and so it is this generalization that we choose to analyze. The most notable of these features is the order of the free data. The free data split into low order and high order pairs. The former pair consists of h(0) and h′ (0) (for simplicity, we will ˆ set h′ (0) = 0), while the latter pair consists of h(n−2) (0) and h(n−1) (0). Note that the trace of B

5

does not vanish identically; rather, the vanishing of the trace imposes the nontrivial condition 1 (1.10) |Eg |2g = A2g 4 when n > 4. If we were to further impose A = 0 as we did when n = 4, the only solutions would be Einstein metrics. Theorem 1.5. Let (M, g) be asymptotically hyperbolic and either even-dimensional, or odddimensional withconformal infinity whose Fefferman-Graham obstruction tensor vanishes. Let 1 2 ′′ ′ ′ g = x2 dx ⊕ hx be Poincar´e-Bach with −(n−2) trh0 h (0) = Scalh0 6= 0 and h (0) := hx x=0 = 0. Let Φ and Ψ be tracefree symmetric (0, 2)-tensors on conformal infinity such that Ψ is h0 divergenceless. Then for each such Φ and Ψ there is a unique formal power series solution of ˆg = 0 such that tf h h(n−2) (0) = Φ, tf h h(n−1) (0) = Ψ. the equations B 0 0 We have not given the most general statement possible. Greater generality would complicate matters and possibly obscure the main points, arguably without significantly enhancing the interest of the theorem. There are three main restrictions, which we now outline. The condition h′ (0) = 0 is imposed mainly for convenience. Without it, some of our expressions become more complicated without compensating gains in insight. We see from the theorem that the remaining data split in a manner reminiscent to that of the Poincar´e-Einstein case, with “Neumann data” now consisting of Φ and Ψ (obviously the condition h′ (0) = 0 is also of Neumann type, but were h′ (0) not fixed it would be natural to think of it as low-order data and pair it with the Dirichlet data h0 ). The gap between the orders of these types of data disappears when n = 4. ˆ = 0 equation that fixes the conformal The condition on Scalh0 is related to the part of the B gauge. This equation is merely quasi-linear, and the Frobenius-type technique used in FeffermanGraham type analyses can fail. It happens not to fail when this condition is met. Finally, if n is odd but the Fefferman-Graham obstruction tensor of the conformal boundary does not vanish, there exist families of formal polyhomogeneous series solutions of the equations Bg = 0 with the same free data as above. But there are no new, further obstructions to formal power series solutions beyond the obstruction in odd bulk dimension already known from the Poincar´e-Einstein case [12, 13] (at least when Scalh0 6= 0). Intuitively one can see this by ˆ is homogeneous in E := Rc +(n − 1)g. We are essentially expanding the Bach noting that B tensor about E = 0, while expanding E about a background hyperbolic metric. The recurrence relation for the coefficients of a linear homogeneous equation, here the equation (or, rather, ˆ to E, can always be solved (though the equation for B ˆ is only system of equations) relating B quasi-linear in E, so issues arise as discussed above). In contrast, the usual Fefferman-Graham obstruction arises because the recurrence relation for the Einstein tensor expansion is about a background metric determined by h0 , not about zero, and so the Poincar´e-Einstein problem is not homogeneous. While we also expand about a background metric determined by h0 , we introduce no new source of nonhomogeneity and, concomitantly, no new obstructions. 1.4. Preview. This paper is organized as follows. In Section 2 we state our conventions and briefly recall the basic theory of asymptotically hyperbolic metrics and Poincar´e-Einstein metˆg in terms of the tensor Eg := Rcg +(n − 1)g. Section rics. In Section 3 we expand Bg and B 4 is dedicated to the case of n = 4. In Section 4.1 we discuss the equation B ⊥ = 0 in n = 4 dimensions, while in Section 4.2 we apply the Bianchi identity and obtain a condition on the divergence of the free data h(3) (0). The equation A = 0 is discussed in Section 4.3. An alternative to fixing the conformal gauge by setting A = 0 is instead to fix the Q-curvature. This is

6

AGHIL ALAEE AND ERIC WOOLGAR

discussed in Section 4.4. The proofs of Theorem 1.1 and Corollary 1.2 then follow quickly from the earlier subsections and are given in Section 4.5. ˆ ⊥ = 0 equation, We discuss the n ≥ 5 case in Section 5. Section 5.1 is dedicated to the B ⋄ ˆ = 0 equation (i.e., the mixed components of B), ˆ and Section 5.3 to the Section 5.2 to the B ˆ B00 = 0 equation where the issue of the conformal gauge arises. With this background, the proof of Theorem 1.5 is then quite brief and comprises Section 5.4. 1.5. Acknowledgements. AA was supported by a post-doctoral fellowship from the Natural Sciences and Engineering Research Council (NSERC). The work of EW was supported by an NSERC Discovery Grant RGPIN 203614. Both authors are grateful to the Fields Institute for Research in Mathematical Sciences, where much of this work was carried out, and to the organizers of its 2017 Thematic Programme on Geometric Analysis for a stimulating environment. We are also grateful to the Banff International Research Station for hosting us at its workshop 18W5108 and to C Robin Graham for discussions and helpful comments on an earlier draft. 2. Preliminaries 2.1. Notation and conventions. As already stated, we use n = dim M to be the dimension of the bulk manifold (M, g). We define the rough (or connection) Laplacian to be the trace of the Hessian, i.e., ∆g := trg Hess = gab ∇a ∇b for a given Levi-Civita connection ∇g . In index notation, we have

(2.1)

Ra bcd = ∂c Γabd − ∂d Γabc + Γace Γebd − Γade Γecd , 1 Wabcd = Rabcd − (gac Rbd − gad Rbc − gbc Rad + gbd Rac ) (n − 2) 1 + (gac gbd − gad gbc ) (n − 1)(n − 1) 1 (gac Ebd − gad Ebc − gbc Ead + gbd Eac ) = Rabcd − (n − 2) A n + (gac gbd − gad gbc ) + (gac gbd − gad gbc ) . (n − 1)(n − 2) (n − 2)

where Rabcd := gae Re bcd and we define

Eab := Rab + (n − 1)gab ,

(2.2)

A := gab Eab = R + n(n − 1) .

We also define the Schouten tensor (2.3)

Pab

1 := (n − 2)

1 Rgab Rab − 2(n − 1)

,

and the tracefree Einstein tensor 1 Rgab . n Finally, in keeping with standard usage, for a function f depending on a defining function x for conformal infinity, we write f ∈ O(xp ) or f ∈ O(xp ) if there are constants C > 0 and ǫ > 0 such that |f | < Cxp for all x < ǫ. Clearly, if f ∈ O(xp ) for some p > q, then f ∈ O(xq ) as well.

(2.4)

Zab := Rab −

7

¯ be a compact manifold-with-boundary 2.2. Asymptotically hyperbolic metrics. Let M with interior M . A metric g on M is called conformally compactifiable if there is a C ∞ metric ¯ and a positive function ρ : M → (0, ∞), such that g¯ on M

g = ρ−2 g¯ ¯ with ρ = 0 and dρ 6= 0 pointwise on ∂M . on M , and such that ρ extends smoothly to M ¯ We refer to ∂ M as the boundary-at-infinity of M . It is sometimes denoted by ∂∞ M . The conformal equivalence class [h] of h := g¯|∂ M¯ is called the conformal boundary of (M, g). We call ¯ ) = 1. ρ a defining function for the conformal boundary. We can always arrange that |dρ|2g¯ (∂ M 1 2 If g¯ is C , we can solve the eikonal differential equation |dx|g¯ = 1 in a collar neighbourhood ¯ , subject to the boundary condition x = 0 on ∂M . Then x is called a special defining of ∂ M function and (M, g) is called conformally compactifiable and asymptotically hyperbolic, or simply asymptotically hyperbolic. On a neighbourhood of conformal infinity, the metric can then be written in the form of equation (1.3); equivalently, dx2 + hx is a metric in Gaussian normal coordinate form, and g is said to be in Graham-Lee normal form. By analyzing the formula for the conformal transformation of the curvature, one then sees that the sectional curvatures of an asymptotically hyperbolic metric approach −1 as x → 0. There is some freedom to choose x, corresponding to the freedom to choose a conformal representative h0 in [h]. We will choose a representative h0 below, so that x will be determined, but the freedom to vary these choices remains. For greater detail, please see [13, 11]. (2.5)

2.3. Poincar´ e-Einstein metrics. These are asymptotically hyperbolic Einstein metrics. They obey the negative Einstein equation (2.6)

Eg := Rcg +(n − 1)g = 0.

on the bulk n-dimensional manifold (M, g). We briefly review the Fefferman-Graham expansion for these metrics. If we insert (1.3) into (2.6), we obtain 1 1 1 2 E00 = − trhx h′′x + trhx h′x + h′x hx 2 2x 4 1 (2.7) E ⋄ = divhx h′x − d trhx h′x 2 1 (n − 2) ′ 1 1 1 ⊥ hx + hx trhx h′x + h′x ◦ h′x − h′x trhx h′x + Rchx Ehx = − h′′x + 2 2x 2x 2 4 ⊥ where E is the tensor on the level sets x = const obtained by projecting E onto the tangent spaces of these sets, E00 = E(∂x , ∂x ), and E ⋄ is the covector field on the levels sets of x defined by E ⋄ (∂yi ) = E(∂x , ∂yi ). If one computes the order-l derivative of the above expression for E ⊥ with respect to x, the result is (2.8)

xhx(l+2) + (l − n + 2)hx(l+1) − hx trhx hx(l+1) = F (hx , . . . , hx(l) ) , l = 0, 1, 2, . . .

Setting x = 0 in this equation allows one to compute by iteration the x-derivatives of order (n−1) 1, . . . , n − 2 of hx at x = 0 in terms of h(0) . When l = n − 2 the coefficient of tf hx hx will vanish. If the tracefree part of F does not vanish under these circumstances, then there is an obstruction to the existence of the Mclaurin expansion of hx about x = 0. The nonvanishing terms define the ambient obstruction tensor which is of much interest in conformal geometry. The obstruction is avoided by adding logarithmic terms so that we no longer have a Mclaurin

8

AGHIL ALAEE AND ERIC WOOLGAR (n−1)

is expansion for hx , but instead have a polyhomogeneous expansion. In any case, tf hx hx free data and can be freely chosen. Once it has been chosen, the iteration can be restarted and continued to all orders, either as a Mclaurin expansion or a polyhomogeneous expansion, as appropriate. The coefficients of the higher order terms in the expansion will in general depend on both h(0) and tf hx h(n−1) (0), but are otherwise completely determined. Two important results (2l+1)

easily derived from this iteration procedure are that (i) all the odd derivatives hx vanish at (n−1) vanishes at x = 0. x = 0 for 2l + 1 < n − 1, and (ii) when n is even, trhx hx Because of the second Bianchi identity, one usually focuses attention on Eh⊥x alone, but the vanishing of Eh⋄x imposes conditions on the divergence of hx which govern the divergence of certain data. Let n be even. Differentiating Eh⋄x with respect to x (n − 2)-times using (2.7), (n−1)

(n−1)

, evaluated at x = 0, is given by a sum of terms each − d trhx hx we obtain that divhx hx (2l+1) , for some l such that 2l + 1 < n − 1. But in the of which has a factor of the form hx x=0 (n−1)

(n−1)

− d trhx hx last paragraph we noted that each odd derivative must vanish. Then divhx hx (n−1) vanishes at x = 0, and since trhx hx itself vanishes at x = 0, we conclude that for even n then (n−1) = 0. In the AdS/CFT correspondence, this allows for the interpretation of divhx tf hx hx x=0 (n−1) as the vacuum expectation value of the CFT stress-energy tensor. The vanishing tf h hx x

x=0 (n−1)

means that there is no conformal anomaly (which would break the conformal of trhx hx (n−1) implies that the appropriate invariance of the CFT), while the vanishing of divhx hx x=0 Ward identity is also anomaly-free. (n−1) in terms of lower derivatives of hx For odd n, this analysis determines divhx tf hx hx x=0 at x = 0, but it need not vanish. Again, for greater detail, please see [13, 11]. 3. The Bach tensor 3.1. Bach tensor in terms of E and W . In this section, we record the main formulas used to expand the Bach tensor in a series. The formulas are straightforward to derive, but the derivations are often tedious and lengthy calculations, so we include only the main intermediate steps in the derivation. To begin, the Bach tensor can be expanded in terms of W , E, and A by Lemma 3.1. Bac

(3.1)

1 (n − 2) ∇a ∇c A − gac ∆A + 2Wdabc E bd ∆Eac − 2(n − 1) 2(n − 1) n A A2 1 b 2 − Eac + Ea Ebc − |E| − gac (n − 2) (n − 1) (n − 2) (n − 1) +nEac − gac A} ,

1 = (n − 2)

where, furthermore, writing (3.2)

g=

1 g˜ , g˜ = dx2 ⊕ hx , x2

and using the coordinate notation xa = (x0 , xi ), i ∈ {1, . . . , n − 1} so that x0 ≡ x, then ∆Eac ≡ ∆g Eac

(3.3)

i h ˜ 0 Eac + 2δa0 ∇ ˜ b Ebc + 2δc0 ∇ ˜ b Eab − 2∇ ˜ a E0c − 2∇ ˜ c E0a − n∇ ˜ 0 Eac = x2 ∆g˜ Eac + x 6∇ − 2(n − 2)Eac + 2˜ gac E00 − nδa0 E0c − nδc0 E0a + 2δa0 δb0 E00 + hij Eij .

9

˜ is the Levi-Civita connection compatible with g˜. Proof. First, we remark on notation. Here ∇ ˜ b Eab := g˜bc ∇ ˜ b Eac . ˜ a Ebc , we mean (∇∂ E) (∂b , ∂c ), and ∇ By ∇ a These results are by direct and simple, if tedious, computation. To obtain (3.1), simply plug (2.6) into (1.1) and compute using the second Bianchi identity. To obtain (3.3), note that the ˜ a of g˜ab are related to those of gab (denoted Γa ) by connection coefficients Γ bc bc

(3.4)

˜ a − 1 δ0 δa + δ0 δa − δa g˜bc . Γabc = Γ bc b c c b 0 x

The usual expansion for a connection in terms of its coefficients yields

(3.5)

˜ a Ebc + 1 2δ0 Ebc + δ0 Eac + δ0 Eab − g˜ab E0c − g˜ac E0b . ∇a Ebc = ∇ a c b x

Now differentiate once more by applying ∇a to (3.5) and use that ∆g Eac = g bd (∇b ∇d Eac ). This is lengthy but simple and we omit the details.

It will be useful to expand equation (3.1) componentwise. The non-vanishing Christoffel symbols of the Levi-Civita connection of g˜ij in the coordinates xa = (x0 = x, xi ) are

(3.6)

˜ 0ij = − 1 h′ij =: Kij , Γ 2 ˜ i0j = Γ ˜ i0j = 1 hik h′ = −hik Kjk =: K i j , Γ jk 2 i i ˜ =Ξ , Γ jk jk

where the Ξijk are the Christoffel symbols of the Levi-Civita connection D = Dx compatible with hx on each constant-x slice. Then we easily compute that

(3.7)

˜ 0 E00 = E ′ , ∇ 00 ˜ i E00 = Di E00 + 2Ki k E0k , ∇ ˜ 0 E0i = E ′ + Ki k E0k , ∇ 0i

˜ i E0j = Di E0j + Ki k Ejk − Kij E00 , ∇ ′ ˜ 0 Eij = Eij ∇ + Ki k Ejk + Kj k Eik , ˜ i Ejk = Di Ejk − Kij E0k − Kik E0j . ∇

10

AGHIL ALAEE AND ERIC WOOLGAR

Differentiating these expressions once more and summing, one obtains the expressions

(3.8) ′′ ′ (∆g E)00 = x2 E00 + ∆h E00 − HE00 − 2|K|2h E00 o +2D i Ki j E0j + 2K ij Di E0j + 2K i j K jk Eik ′ − x (n − 6)E00 − 4D i E0i − 4K ij Eij + 4HE00 − 4(n − 2)E00 + 2hij Eij , n ′′ ′ ′ + ∆h E0i − HE0i + 2Ki j E0j + (Ki j )′ E0j − 2Ki j Kj k E0k − |K|2h E0i (∆g E)0i = x2 E0i o −HKi j E0j + 2K jk Dj Eik + (Dj K jk )Eik − 2Ki j Dj E00 − (D j Eij )E00 ′ − x (n − 6)E0i + nKi j E0j − 2D j Eij + 2HE0i + 2Di E00 − (3n − 4)E0i , n ′ ′ ′ ′′ + 2Ki k Ejk + 2Kj k Eik − HKi k Ejk − HKj k Eik + ∆h Eij − HEab (∆g E)ij = x2 Eij

+(Ki k )′ Ejk + (Kj k )′ Eik + 2Ki k Kj l Ekl − 2Kik D k E0j − 2Kjk D k E0i o −(D k Kik )E0j − (D k Kjk )E0i + 2Ki k Kjk E00 o n ′ − x (n − 6)Eij + (n − 8) Ki k Ejk + Kj k Eik − 2 (Di E0j + Dj E0i ) − 4Kij E00

− 2(n − 2)Eij + 2hij E00 .

In the above, indices are raised with h−1 , denoted as usual by hij . We need also that

(Hess A)00 = A′′ +

1 ′ A , x

(Hess A)0i = Di (A′ ) + Ki k Dk A +

(3.9)

1 Di A , x

1 (Hess A)ij = Di Dj A − Kij A′ − hij A′ , x =⇒ ∆g A = x2 A′′ − HA′ + ∆h A − (n − 2)xA′ ,

where (2.7) yields

(3.10)

2 x2 3 2 A = − x2 trhx h′′x + (n − 1)x trhx h′x + x2 h′x hx − trhx h′x + x2 Scalhx 4 4 = x2 2H ′ − |K|2hx − H 2 + Scalhx − 2(n − 1)xH .

11

Putting this all together, we have

B00

(3.11)

x2 = (n − 2)

(2n − 3) 1 1 1 ′′ ′ ′ E00 ′′ − trhx (E ⊥ ) − HE00 + H trhx (E ⊥ ) 2 2 2(n − 1) 2(n − 1) 1 ′ ∆h A˜ + 2Di (K ij E0j ) + 2K ij Di E0j −2K ij Eij − (K ij )′ Eij + ∆h E00 − 2(n − 1) 1 2 +2K ij Ki k Ejk + − |E0i |2h HK ij Eij − 2|K|2h E00 − E00 (n − 1) 2 1 1 1 ⊥ ⊥ ⊥ 2 trhx E E00 trhx E + |E |h − + (n − 1) (n − 2) (n − 1) x (6n − 7) ′ ′ + HE00 −(n − 4)E00 − 2 trhx (E ⊥ ) + 4D j E0j − (n − 2) (n − 1) 1 H trhx E ⊥ + (n − 1) 2x4 W0i0j hik hjl Ekl , − 3E00 + (n − 2)

where we write A˜ := trg˜ E ∈ O(1/x). Continuing, we have

x2 B0i = (n − 2)

(3.12)

i h (n − 2) ′ ′ ′′ ′ ′ + trhx (E ⊥ ) + ∆h E0i Di E00 E0i + 2Ki j E0j − HE0i − 2(n − 1) (n − 2) h +D k Kk j Eij − Kik E00 + K jk Dj Eik − Ki j Dj E00 − 2Di (K jk Ejk ) 2(n − 1) i ′ jk h E0k − HKi j E0j − |K|2h E0i +Ki j Dj E00 + Ki j Dj (trhx E ⊥ ) + Kij h i n n ⊥ jk E00 E0i + E0j h Eki + E00 + trhx E E0i − (n − 2) (n − 1)(n − 2) 3(n − 2) x ′ Di E00 + trhx E ⊥ −(n − 6)E0i − 2Di E00 + 2D j Eij − + (n − 2) 2(n − 1) −nKi j E0j − 4HE0i 2x4 − 2E0i + W0jik hjl hkp Elp − W0j0i hjk E0k , (n − 2)

12

AGHIL ALAEE AND ERIC WOOLGAR

and finally Bij =

(3.13)

i h 1 x2 ′′ ′ ′ ′ ′′ ′′ + 2Ki k Ejk + 2Kj k Eik E00 + trhx (E ⊥ ) hij − HEij Eij − (n − 2) 2(n − 1) 1 1 ′ ′ + [(n − 2)Kij + Hhij ] E00 + [(n − 2)Kij + Hhij ] trhx (E ⊥ ) 2(n − 1) 2(n − 1) n 1 2 2 ⊥ 2 kl ′ kl E00 + |E |h hij K Ekl hij − Eik h Ejl − − (n − 1) (n − 2) n i h n 1 1 ⊥ ⊥ + E00 + trhx E Eij − E00 hij − (trhx E )hij (n − 1)(n − 2) n n n x ′ −(n − 6)Eij − (n − 4) Ki k Ejk + Kj k Eik + 4Kij E00 − 2HEij + (n − 2) i 2(n − 4) (n − 4) h ′ ′ + E00 hij + trhx (E ⊥ )hij + K kl Ekl hij (n − 1) (n − 1) i h 1 ⊥ [(n − 2)Kij + Hhij ] E00 + trhx E + (n − 1) (n − 4) 1 3 (trhx E ⊥ )hij + E00 hij − Eij − (n − 2) (n − 1) (n − 1) i x4 h Wikjl hkp hlq Epq + 2Wikj0 hkl E0l + Wi0j0 E00 . + (n − 2)

Despite their lengths, the above expressions have a simple structure, owing at least in part to the quasilinearity of the Bach tensor as a function of the metric. For example, the tensor E of an asymptotically hyperbolic metric is O(x) in all dimensions, while its normal form components can diverge as O(1/x). For the Bach tensor we can now read off from the above expressions the comparable result. Lemma 3.2. The normal-form components of the Bach tensor of an asymptotically hyperbolic n-manifold admit a C 0 extension to conformal infinity, and vanish at conformal infinity when n = 4. We have ( 2 ˆ g ∈ O(x ), n > 4 . (3.14) |B| O(x3 ), n = 4 Proof. In deriving equations (3.11), (3.12), and (3.13), we have not expanded the Weyl tensor contribution to the Bach tensor. To establish the lemma, such an expansion is not necessary. g To see this, observe first that the conformal properties of the Weyl tensor are such that Wabcd = a g˜ 1 g ˜ g ˜ bcd ∈ O(1), so the components of W with respect to a normal-form basis x2 Wabcd . Now W g˜ g {∂0 , ∂i } obey Wabcd ∈ O(1) as well (lowering the index with g˜). Hence Wabcd ∈ O(x−2 ). Further, ij h ∈ O(1) and, by equations (2.7), Eab ∈ O(1/x). Thus, any product of the form W g ∗ h−1 ∗ E g or W g ∗ h−1 ∗ h−1 ∗ E g (with all indices lowered in W ) has components that lie in (at worst) O(x−3 ). But each time such a term appears in equations (3.11), (3.12), and (3.13), it appears with coefficient x4 , and hence the components of these terms in the normal-form basis vanish at least as O(x). We simply substitute equations (2.7) into equations (3.11)–(3.13). Together with the fact that the Weyl tensor term Wdabc E bd in (3.1), expressed in a normal form basis, is in O(x),

13

straightforward cancellation of terms now leads directly to B00 ∈ O(x) ,

B0i ∈ O(x) ,

2 1 ′ 2 ′ H hij + |K|h hij (n − 3) Kij − (3.15) (n − 1) (n − 1) 1 1 h +(n − 4) Ki k Kjk − |K|2h hij − Zij H 2 hij + HKij − (n − 1) (n − 1) + O(x) , where we write h for the boundary metric h := hx x=0 and write Z h := tf hx Rch . Thus when n = 4, the normal-form components of B are of order x. If the components of a (0, 2)-tensor in the normal-form basis are in O(xp ), obviously the tensor norm of that tensor is ˆ g ∈ O(x3 ). in O(xp+2 ), so |B|g = |B| For n > 4, we must compute the additional terms on the right-hand side of (1.9). Using (2.7), (3.10), and (3.15) we find that ˆ00 = B00 − 1 (n − 4) |K|2 − H 2 + O(x) B h 2 1 = − (n − 4) |K|2h − H 2 + O(x) , 2 ˆ0i = O(x) , B 1 (3.16) 2 2 k ˆ |K|h − H hij Bij = Bij + (n − 4) Ki Kjk − HKij − 2 (n − 3) 1 ′ = (n − 4) H ′ hij + 2Ki k Kjk − HKij Kij − (n − 2) (n − 1) n 1 2 2 h H hij − Zij + O(x) . − |K|h hij + 2 2(n − 1) (n − 4) Bij = (n − 2)

ˆ00 and B ˆij are of order 1, and so |B| ˆ g ∈ O(x2 ) as claimed. Hence B

ˆ⋄ 3.2. The Bianchi identity. In the sequel we will not make direct use of the expansions of B ˆ0i ) and B ˆ00 , except somewhat in section 5.3. Instead, we will solve the equation B ˆ⊥ = 0 (i.e., B ˆ to show that the remaining components vanish. and use the vanishing of the divergence of B ˆ Indeed, when n = 4 (so B = B), B00 will vanish simply because then the Bach tensor is traceless. The vanishing of the divergence (Proposition 1.4) yields the equations (n − 2) ′ ˆ00 + D j B ˆ0j = K jk B ˆjk , ˆ00 − H + (3.17) B B x (n − 2) ˆ ′ ˆ ˆij . B0i − H + B0i = −D j B (3.18) x ˆij = 0 this becomes a homogeneous linear system for (B ˆ00 , B ˆ0i ), admitting Obviously when B the trivial solution. ˆ (α) (0) = 0 for all 0 ≤ α ≤ β. If β ≥ n − 2 assume further Proposition 3.3. Assume that B ij (n−2) (β+1) ˆ ˆ that B0i (0) = 0. Then B0i (0) = 0. In addition, under these conditions we also have (β+1) ˆ ˆ (n−2) (0) = 0. B (0) = 0 provided that, for β ≥ n − 2, we assume as well that B 00

00

14

AGHIL ALAEE AND ERIC WOOLGAR

ˆ0i ∈ O(x) and B ˆij ∈ O(1). Then expand B0i = Proof. We have from Lemma 3.2 that B ∞ ∞ ∞ P P P ˆ (n−2) (0) = 0 ⇔ bi(n−2) = 0. bi(β) xβ , H = h(β) xβ , and D j Bij = ci(β) xβ . Note that B 0i

β=1

β=0

β=2

Then it is an easy exercise to expand (3.18) and obtain ∞ X

β=1

(3.19) =⇒

∞ X

β=0

[β − (n − 2)] x

β−1

[β − (n − 3)] xβ −

−

β ∞ X X β=1

α=1

β ∞ X X

β=1

bi(α) h(β−α)

bi(α) h(β−α)

α=1

!

Equating coefficients of powers of x, we have

bi(1) = − (3.20)

!

β

x −

xβ −

∞ X

ci(β) xβ = 0

β=0

∞ X

ci(β) xβ = 0 .

β=0

1 c , (n − 3) i(0)

[β − (n − 3)] bi(β+1) = ci(β) +

β X

bi(α) h(β−α) .

α

ˆ (β+1) (0) = 0. One sees from the left-hand side It follows by induction that bi(β+1) = 0 and so B 0i ˆ (n−2) (0) = 0 of (3.20) that the induction pauses when β = n − 3, but then the assumption B 0i fulfils the inductive hypothesis and the induction can be restarted and continued arbitrarily. (β+1) The proof for B00 now follows similarly. ˆ ≡ trg B = 0. From this, we see that B00 vanishes We recall that for n = 4, we have that trg B order-by-order whenever Bij does. We do not need to appeal to the above proposition. 4. Four dimensions ˆ notation in this section since B ≡ B ˆ 4.1. The equation for Bij . We set n = 4 and drop the B for n = 4. Then equation (3.13) becomes i 1 2 1 h ′′ ′′ ′ ′ ′ ′′ + 2Ki k Ejk + 2Kj k Eik E00 + trhx (E ⊥ ) hij − HEij Bij = x Eij − 2 6 1 1 1 1 ′ ′ + Kij + Hhij E00 + Kij + Hhij trhx (E ⊥ ) 3 2 3 2 1 2 kl ′ 2 E00 + |E ⊥ |2h hij − K Ekl hij − 2Eik hkl Ejl + 3 2 (4.1) i 2h 1 1 ⊥ ⊥ + E00 + trhx E Eij − E00 hij − (trhx E )hij 3 4 4 h i 2 1 1 ′ ⊥ + x 2Eij + 4Kij E00 − 2HEij + Kij + Hhij E00 + trhx E 2 3 2 i h 1 + E00 hij + x4 Wikjl hkp hlq Epq + 2Wikj0 hkl E0l + Wi0j0 E00 . 2 In view of Lemma 3.2, the above expression can be expanded as a power series in x. If one substitutes (2.7) into (4.1), one obtains an expression that is perfectly regular at x = 0—indeed, with vanishing constant term—despite the fact that the expression for E in (2.7) has some divisions by x. In particular, let LWT denote a sum of lower weight terms. These are terms that are regular at x = 0 and have the form of a (possibly) derivative-dependent coefficient

15 (p)

C(hx , h′x , . . . , hx ) multiplying a nonnegative power of x, say xq . The weight is defined to be the order of the highest x-derivative of hx appearing in C minus the power of x multiplying (4) the term; i.e., the weight is p − q. For example, the weight of the term − 14 x2 tf hx hij (x) is 4 − 2 = 2, while a term such as x2 (trhx h′ ) tf hx h′ij would have weight 1 − 2 = −1. Then we have the following. Lemma 4.1. 1 (4) Bij (x) = − x2 tf hx hij (x) + LWT . 4 Proof. Simply plug (2.7) into (4.1). While the resulting expression is very lengthly, one can eliminate most terms immediately by observing that the highest weight contributions must arise ′′ , xE ′ , and E h . Expanding these terms using (2.7) yields the from the linear terms − 21 x2 Eij 00 ij ij result. (4.2)

Lemma 4.2. Let Bij (x) = 0 and n = 4. Then for some h0 -tracefree tensor F on ∂∞ M and any s ≥ 4 we may write (s)

tf h0 hij (0) = Fij (h0 , h′ (0), . . . , h(s−1) (0)) .

(4.3)

Proof. Equation (4.2) with Bij (x) = 0 implies that 1 2 (4) x tf hx hij (x) = LWT . 4 If one differentiates the left-hand side r-times, with r ≥ 2, and sets x = 0, one obtains 14 r(r − 1) tf h0 h(r+2) (0) plus terms of lower differential order. On the right-hand side, consider a term of weight w := p − q for p and q as described immediately before Lemma 4.1. If r < q, a factor of x remains after differentiation, so the term vanishes when we set x = 0. Hence take r ≥ q. Then the term contributes as (p) ∂ r−q r! ′ (r−q)! ∂xr−q x=0 C(hx , hx , . . . , hx ). Thus, the highest derivative that can arise from this term is

(4.4)

h(r−q+p) (0). Now r − q + p = r + w < r + 2 since w < 2. Combining both sides, we have that tf h0 h(r+2) (0) equals a sum of terms that depend on no derivative higher than h(r+1) (0). Now set s = r + 2. This lemma does not determine the trace of h(r) (0) for any order r. It does, however, show that one can determine all the coefficients in a formal power series solution of Bij = 0 in the case of an asymptotically hyperbolic 4-dimensional bulk manifold in terms of given data h0 ≡ h(0), h′ (0), h′′ (0), and h′′′ (0) at the conformal boundary, if one is also given as data the traces trh0 h(r) (0) for all r. There are no obstructions, so it is not necessary to augment the power series with logarithmic terms. Corollary 4.3. If Bij (x) = 0 then there is an h0 -tracefree tensor G on ∂∞ M such that (4.5)

(s)

tf h0 hij (0) = G(h0 , h′ (0), h′′ (0), h′′′ (0), trh0 h(4) (0), . . . , trh0 h(s−1) (0)) , s ≥ 4 .

Proof. Immediate from Lemma 4.2 by induction on s.

4.2. The equation for B0i . We know from Proposition 3.3 that, in the presence of the condition Bij (x) = 0, then the series expansion B0i (x) is determined by the divergence identity except for the coefficient of the order xn−2 term. Here we have n = 4. Then the only additional information ′′ (0) to be learned from solving the B0i (x) = 0 equation directly is the condition(s) under which B0i

16

AGHIL ALAEE AND ERIC WOOLGAR

′′ (0) = 0 imposes a condition on the data will vanish. The next result shows that the condition B0i h(3) (0) which, in the Poincar´e-Einstein setting, has an important interpretation in the AdS/CFT correspondence. ′′ (0) = 0 ⇔ div tf (3) Proposition 4.4. Let n = 4 and choose h′ij (0) = 0. Then B0i h0 h0 h (0) = 0.

Proof. Explicit computation beginning with (3.12) and using (2.7) yields 1 1 2 (3) ′′ x divhx tf hx hx − (n − 4)x divhx tf hx hx + LWT . (4.6) B0i = 2 2 i

Now set n = 4, differentiate twice with respect to x, and set x = 0. Upon taking two xderivatives of (3.12), one can see by inspection (using as well (2.7)) that each term arising from twice differentiating the terms denoted LWT either contains a factor of Kij or has a coefficient of x or x2 . Hence these terms vanish upon setting x = 0 and then Kij := − 21 h′ij (0) = 0. Thus we obtain (3) ′′ (4.7) 0 = B0i (0) = D k tf hx hik (0) .

The same result in the Poincar´e-Einstein case is essential for the AdS/CFT correspondence, because it allows for the interpretation of h(3) (0) (or, for a 2n-dimensional bulk, h(2n−1) (0)) as the vacuum expectation value of the stress-energy for a conformal field theory defined on ∂∞ M . 4.3. The condition A = 0. This condition is imposed in Corollary 1.2 and in Theorem 1.5. Its role is to choose a unique conformal representative within a conformal class [g] of solutions of B = 0. We may compute from (2.7) that 2 3 x2 A := trg E = −x2 trhx h′′x + (n − 1)x trhx h′x + x2 |h′x |2hx − trhx h′x + x2 Scalhx . 4 4 It is convenient to write the condition A(x) = 0 as 2 3 x (4.9) 0 = −x trhx h′′x + (n − 1) trhx h′x + x|h′x |2hx − trhx h′x + x Scalhx , 4 4 from which it follows immediately that

(4.8)

trh0 h′ (0) = 0 .

(4.10)

If we further assume that h′ (0) = 0, then we can differentiate (4.9) once and set x = 0 to obtain (4.11)

trh0 h′′ (0) = −

1 Scalh0 ≡ −2 trh0 Ph0 . (n − 2)

If one differentiates (4.9) twice, sets x = 0, and uses h′ (0) = 0, then one obtains (4.12)

trh0 h′′′ (0) = 0 .

In general, if one differentiates (4.9) r ≥ 1 times with respect to x and sets x = 0, one obtains (4.13)

0 = (n − 1 − r) trh0 h(r+1) (0) + Fn (h0 , h′ (0), . . . , h(r) (0))

for some function Fn that depends on the dimension n. When r = n − 1, one see from this that trh0 h(n) (0) is undetermined, and that there are no solutions unless Fn (h0 , h′ (0), . . . , h(n−1) (0)) = 0 as well.

17

Proposition 4.5. For n = 4, choose tf h0 h′ (0) = 0, tf h0 h′′ (0) = −2Zh0 , tf h0 h′′′ (0) = Ψ for Ψ as in Theorem 1.1, and trh0 h(4) (0) = T4 for some arbitrary function T4 . If Bg = 0 and Ag = 0, then trh0 h(k) (0) is uniquely determined for all k. Proof. This is obvious from the above expressions (4.10)–(4.13) and Theorem 1.1, provided equation (4.13) has a solution; i.e., provided F4 = 0. With the chosen data, we have from (4.10)–(4.12)) that (4.14) Then we obtain (4.15)

h′ (0) = 0 , h′′ (0) = −2Ph0 , tf h0 h′′′ (0) = Ψ .

h i F4 = −6 |Ph0 |2h0 − 6 (trh0 Ph0 ))2 + 6 Rhij0 (Ph0 )ij − D i D j (Ph0 )ij + ∆h0 (trh0 Ph0 )) .

The Bianchi identity ensures that −D i D j (Ph0 )ij + ∆h0 (trh0 Ph0 )) = 0, and it is a simple matter to check that the first three terms on the right of (4.15) sum to zero as well, so F4 vanishes as claimed. This is, of course, not an accident. The conditions h′ (0) = 0, h′′ (0) = −2Ph0 , trh0 h′′′ (0) = 0, imply that g is a 4-dimensional APE (asymptotically Poincar´e-Einstein) metric. There is no obstruction to power series in x for such metrics when the bulk dimension n is even, meaning that for this data the Einstein equations can be solved to order n inclusive (and indeed to any order in x). Therefore, we can always solve the equation A = 0 to order n inclusive (for n even), given data for an APE metric. Beyond order n the coefficient on the left-hand side of equation (4.13) never vanishes, so no obstruction to a recursive solution arises. 4.4. The condition Q − 6 = 0. Rather than fixing A = 0, we can fix the Q-curvature. We recall that the 4-dimensional Q-curvature is 1 Q := −∆g Scalg + Scal2g −3| Rcg |2g 6 1 1 1 (4.16) = − ∆g Ag + (Ag − 12)2 − |Eg − 3g|2g 6 6 2 1 1 1 = − ∆g Ag − Ag − |Eg |2g + A2g + 6 , 6 2 6 so Einstein 4-metrics have Q = 6. This motivates us to consider replacing the condition A = 0 by the condition Q − 6 = 0. Using (3.10) and the last line of (3.9) (and using (2.7) to observe that the |E|2 term is of lower weight), we may rewrite the condition Q − 6 = 0 as 2 2 1 3 2 2 1 4 ′ (3) ′′ 3 (4.17) x trhx h(4) x − x trhx hx + x trhx hx − 2x trhx hx = LWT = x Scalh0 +O(x ) . 6 6 3 3 Differentiating once and setting x = 0, we immediately see that trh0 h′ (0) = 0, which is the same result as we obtained by setting A = 0. As before, set the free data tf h0 h′ (0) to vanish as well, so that h′ (0) = 0. Then we can differentiate (4.17) twice and set x = 0 to obtain trh0 h′′ (0) = − 12 Scalh0 , which is the same condition as arises from setting A = 0, see (4.11). Since tf h0 h′′ (0) is free data for the equation B = 0 as well as for the equation Q − 6 = 0, let us now choose tf h0 h′′ (0) = Zh0 , so that h′′ (0) = −2Ph0 where Ph0 is the Schouten tensor of h0 . That is, we choose data that correspond to Poincar´e-Einstein metrics to order x2 inclusive. One can now compute the coefficients of the higher-order terms in (4.17). The coefficient of the x3 term vanishes, hence trh0 h′′′ (0) = 0. (Again, the tracefree part is free data for B = 0 as well as for Q − 6 = 0.)

18

AGHIL ALAEE AND ERIC WOOLGAR

To go to order x4 and beyond, differentiate (4.17) k ≥ 4 times, setting k = 0, and using the choices and results listed in the last paragraph, we now find ( F (h0 ), k = 4, 1 k(k − 4) k2 − 3k + 6 trh0 h(k) (0) = (4.18) (5) (k−1) 6 F (h0 , h (0), . . . , h (0)), k ≥ 5.

From the left-hand side of (4.18), we see that trh0 h(4) (0) is not determined by the condition Q − 6 = 0. But given h0 and the above choices trh0 h′ (0) = 0 and tf h0 h′′ (0) = tf h0 Ph0 ≡ Zh0 , if we also choose values for tf h0 h(3) (0) and trh0 h(4) (0) then all higher-order traces are determined by recursive application of (4.18). Now since the left-hand side of (4.18) vanishes when k = 4, we observe that we must have F (h0 ) = 0 on the right-hand side. But F (h0 ) can be separately computed explicitly. We have done so and find that F (h0 ) = 0. This can also be seen without the explicit calculation, by the following argument. The chosen data and the datum trh0 h(3) (0) = 0 together imply that g is asymptotically Poincar´e-Einstein (APE). Any APE 4-metric has |E| ∈ O(x4 ) [5] and A ∈ O(x5 ) ([21], or simply refer to the preceding subsection). Hence these data alone guarantee that Q − 6 ∈ O(x5 ) so the fourth-order Taylor coefficient in the expansion of Q vanishes. But this coefficient is F (h0 ) (times a non-zero constant). Thus we have shown the following. Proposition 4.6. For n = 4, choose tf h0 h′ (0) = 0, tf h0 h′′ (0) = −2Zh0 , and tf h0 h′′′ (0) = Ψ and trh0 h(4) (0) = T4 for some arbitrary function T4 . If Bg = 0 and Qg − 6 = 0, then trh0 h(k) is uniquely determined for all k. 4.5. The main theorems for n = 4. We now have assembled everything we need to prove the n = 4 results quoted in the Introduction. Proof of Theorem 1.1. Corollary 4.3 allows us to compute iteratively and uniquely the tracefree parts of h(k) (0) for k ≥ 4 in terms of a boundary metric h(0) = h0 , arbitrary data h′ (0), h′′ (0), h′′′ (0), and the traces Ti := trh0 h(i) (0) for 4 ≤ i < k. Proposition 4.4 imposes one restriction on the data, namely that divh0 tf h0 h(3) (0) = 0. Proof of Corollary 1.2. By [12, 13], given h0 there is a unique (formal series expansion for a) Poincar´e-Einstein metric in normal form (1.3) with h(0) = h0 , such that h′ (0) = 0, h′′ (0) = −2Ph0 , h′′′ (0) = Ψ, and trh0 h(4) (0) = (3!) |Ph0 |2h0 as well. But every Poincar´e-Einstein metric is Poincar´e-Bach, so this formal series represents a Poincar´e-Bach metric. Now Proposition 4.5 and Corollary 4.3 allow us to compute iteratively and uniquely both the trace and tracefree parts of h(k) (0) for k ≥ 4 in terms of a boundary metric h(0) = h0 and arbitrary data tf h0 h′ (0), tf h0 h′′ (0), tf h0 h′′′ (0), and T4 := trh0 h(4) (0), yielding a unique formal power series for hx in (1.1). Choose the data so that tf h0 h′ (0) = 0, tf h0 h′′ (0) = −2Zh0 , tf h0 h(3) (0) = Ψ, and trh0 h(4) (0) = (3!) |Ph0 |2h0 . In particular, this series will have h′ (0) = 0, h′′ (0) = −2Ph0 , and h′′′ (0) = Ψ. Thus the free data for the series that solves B = 0 agrees with the corresponding coefficients of a unique Poincar´e-Einstein metric. Since the data uniquely determine the full series, and since there exists a formal Poincar´e-Einstein metric with these data, the formal solution of B = 0 determined by these data must be Poincar´e-Einstein. Proof of Corollary 1.3. The proof is the same except that it relies Proposition 4.6 rather than Proposition 4.5.

19

5. Higher dimensions ˆ ⊥ . We return to equation (3.13). The coefficients of (n − 4) in front of 5.1. Expansion of B several terms no longer cause these terms to vanish, but the highest weight terms are still very simple to extract. We obtain Lemma 5.1. The components of B ⊥ are given by (5.1)

ˆij = − B

1 (n − 4) (n − 4)(n − 3) x2 tf hx h(4) x tf hx h(3) tf hx h′′x + LWT . x − x + 2(n − 2) (n − 2) 2(n − 2)

Proof. By inserting equations (2.7) into (3.13) and counting weights, we see that (5.1) holds ˆij . But from (1.9) we see that the difference between B and with Bij on the left rather than B ˆ consists entirely of lower weight terms. B ˆij = 0 on the left of (5.1) and take derivatives with Then, as with the n = 4 case, we set B respect to x. This yields: ˆij (x) = 0 and s ≥ 2. Then there is a tracefree symmetric (0, 2)-tensor F Lemma 5.2. Let B such that (5.2)

(s)

(s − n + 2)(s − n + 1) tf h0 hij (0) = Fij (h0 , h′ (0), . . . , h(s−1) (0)) .

ˆij (x) = 0 in (5.1) and multiply the equation by 2(n − 2) to remove a denominator Proof. Set B (5.3)

′′ (3) 0 = −x2 tf hx h(4) x + 2(n − 4)x tf hx hx − (n − 4)(n − 3) tf hx hx + LWT .

Setting x = 0 in (5.3), we obtain 0 = −(n − 4)(n − 3) tf h0 h(2) (0) + LWT x=0 . But at x = 0 the only lower weight terms that can contribute are those that depend only on h0 and h′ (0). This yields (5.2) with s = 2. Now differentiate (5.3) s times with respect to x and set x = 0 (and of course set B = 0). Observe that s derivatives of a lower weight term will still be of lower weight (s) than tf hx hx , and so cannot contain a term with s or more x-derivatives of hx unless multiplied by a positive power of x. As a result, the non-zero contributions to the lower weight terms that survive when we set x = 0 will have at most (s − 1) x-derivatives. The F in the above lemma is not meant to be the same one as in Lemma 4.2; it changes with dimension. We will also use F to denote distinct functions (more precisely, distinct sections of tensor bundles, with different numbers of functional arguments) below. The analogue of Corollary 4.3 for n ≥ 5 is tedious to write out but its content is straightforward. It says that the tracefree parts of the coefficients h(k) (0) are either free data or functions of lower order free data. ˆij (x) = 0 in normal form (1.3) Corollary 5.3. For n = dim M ≥ 5, let g be a solution of B ′ (n−2) with hx given by a formal power series. Then h0 , h (0), h (0), and h(n−1) (0) are free data (r) for the series, as are the traces trh0 h for all r. For those s for which tf h0 h(s) (0) is not free data, we have as follows:

20

AGHIL ALAEE AND ERIC WOOLGAR

a) For n = 5,

(5.4)

F (h0 , h′ (0)), F (h , h′ (0), tr h′′ (0), h(3) (0), h(4) (0)), 0 h0 (s) tf h0 h (0) = F (h0 , h′ (0), trh0 h′′ (0), h(3) (0), h(4) (0), trh0 h(5) (0), . . . , tr h(s−1) (0)), h0

s=2, s=5,

s≥6.

b) For n = 6,

(5.5)

tf h0 h(s) (0) =

c) For n ≥ 7,

(5.6) tf h0 h(s) (0) =

F (h0 , h′ (0)), F (h0 , h′ (0), trh0 h′′ (0)),

F (h0 , h′ (0), trh0 h′′ (0), trh0 h(3) (0), h(4) (0), h(5) (0)), F (h0 , h′ (0), trh0 h′′ (0), trh0 h(3) (0), h(4) (0), h(5) (0), trh0 h(6) (0), . . . , trh0 h(s−1) (0)),

F (h0 , h′ (0)), F (h , h′ (0), tr 0

s=2 , s=3 , s=6 ,

s≥7 . s=2,

h0

h(2) (0), . . . , trh0 h(s−1) (0)),

F (h0 , h′ (0), trh0 h(2) (0), . . . , trh0 h(n−3) (0), h(n−2) (0), h(n−1) (0), trh0 h(n) (0), . . . , trh0 h(s−1) (0)),

3 ≤ s < n−2

s≥n.

2 Zh0 where Zh0 := tf h0 Rch0 . In all cases, when s = 2 and h′ (0) = 0 we have tf h0 h′′ (0) = − (n−3)

Proof. The assumption that the series solution exists implies that when s = n − 2 or s = n − 1, equation (5.2) still holds; i.e., for those x-values, the right-hand side Fij is assumed to vanish (see below). By way of example, we consider the simplest case, that of n = 5. For s = 2, the result is immediate from (5.2). Likewise, we see immediately that the left-hand side of (5.2) vanishes for s = 3 and s = 4, so we cannot use this equation to determine h(3) (0) or h(4) (0). For s = 5, the result is again immediate from (5.2) except that we can omit the h0 -tracefree part of h′′ (0) from the list of arguments because we have already shown that tf h0 h′′ (0) is determined by h0 and h′ (0). For s = 6, we can use the s = 5 result to omit tf h0 h(5) (0) from the list of arguments, but trh0 h(5) (0) has not been determined and so must be included. By a finite induction, we see that we can then omit the tracefree parts of h(k) for all 5 ≤ k < s, but not the trace parts. The arguments for n = 6 and n ≥ 7 are similar. The final statement is easily seen by direct calculation from (3.13). Indeed, by inspection one can observe that the only nonvanishing tracefree contributions when x = 0 and h′ (0) = 0 must be terms proportional to tf h0 h′′ (0) coming from E ′ and E ′′ , and a Zh0 term arising from the

21

tracefree part of E ⊥ (see the penultimate line of (3.13)). A detailed calculation establishes the correct proportions of each such term, yielding the coefficient of tf h0 h(k) (0) for each k. The free data are those tf h0 h(k) (0) for which this coefficient vanishes. These results do not discuss the cases s = n − 1 and s = n − 2. For these cases, the left-hand side of (5.2) vanishes, so the corresponding derivatives cannot be determined. However, it is not clear that the right-hand sides vanish. This is the question of obstructions to formal power series solutions. By answering this question, which we will do in the proof of Theorem 1.5, we can remove the assumption that h be given by a formal power series in Corollary 5.3, since the absence of obstructions means that such a formal series exists. ˆ ⋄ . The divergence identity ∇b B ˆab = 0 implies that 5.2. Expansion of B (n − 2) ˆ ′ ˆ ˆij , (5.7) B0i − H + B0i = −D j B x where the right-hand side is the component expression for divhx B ⊥ . This expression admits the zero solution when B ⊥ = 0. Indeed, expanding in a power series and using from (3.16) that ∞ ˆ0i ∈ O(x) so that B ˆ0i = P b(p) xp , we have B p=1

(5.8)

∞ X p=1

i

(p)

[p − (n − 3)] bi xp−1 −

(p)

∞ X p=1

p ∞ X X (p) (q) ci xp−1 , bi h(p−q) xp = p=1

q=1

where h(q) and ci denote the coefficients in the power series expansions of the mean curvature ˆij of B, ˆ respectively. We obtain H or constant-x hypersurfaces and the divergence D i B (1)

bi (5.9)

(p+1)

[p − (n − 3)]bi

1 (0) ci , (n − 3) p X (q) (p) bi h(p−q) , p ≥ 1 , = ci +

=−

q=1

(p)

ˆ0i vanish up to order n − 3 inclusive whenever the and in particular the coefficients bi of B (p) ˆij do. At order n − 2, b(n−2) is free data but if it is chosen to vanish, and coefficients ci of D j B i (p) (p) if the coefficients ci continue to vanish for higher orders, then the coefficients bi continue to vanish for higher orders as well. ˆ0i = 0 is consistent, order-by-order in the power series expansion, with Hence the condition B ˆ the vanishing of Bij , provided that the right-hand side of (5.9) vanishes when p = n − 3. This imposes one condition on the otherwise-free data. To obtain this condition, we will work directly from (3.12) and (1.9) to write h i 1 ′′ j ˆ0i = B x2 D j tf hx h′′′ h − (n − 4)xD tf (5.10) h x ij x ij + LWT . x 2(n − 2) ˆ0i = 0, we obtain Differentiating k ≥ 1 times and setting x = 0 and B (5.11) k[k − (n − 3)]D j tf hx h(k+1) (0) = LWT , k ≥ 1 . ij

We observe that D j

tf hx h(n−2) (0) ij is therefore undetermined but:

22

AGHIL ALAEE AND ERIC WOOLGAR

Lemma 5.4. D j tf hx h(n−1) (0) ij is determined by this equation by terms containing only lowerorder x-derivatives. Proof. Set k = n − 2 in (5.11) and note that a term of weight lower than k + 1 which is non-zero when x = 0 cannot contain an x-derivative of order greater than k. ˆ00 = 0. The divergence identity for B ˆ implies that 5.3. The condition B ˆ00 − H B ˆ00 + D i B ˆ0i + K ij Bij = 0 . ˆ ′ − (n − 2) B B 00 x ∞ ∞ ∞ ˆ00 = P b(p) xp , H = P h(p) xp , D i B ˆ0i + K ij Bij = P c(p) xp , we obtain Using the expansions B (5.12)

p=0

p=0

p=0

the recursive relation

b(0) = 0 , (5.13)

[p − (n − 2)]b(p) = c(p−1) +

∞ X q=0

b(q) h(p−q−1) , p ≥ 1 .

ˆ00 are determined by the low-order data, and vanish Thus for p < n−2, the coefficients b(n−2) of B ˆ ˆ when the coefficients of B0i and Bij vanish to that order. The coefficient b(n−2) is undetermined ˆ0i and is therefore part of the free data but if it vanishes and the higher-order coefficients of B ˆ ˆ and Bij vanish up to some given order then the coefficients of B00 continue to vanish to that order as well. The left-hand side of the second equation in (5.13) is identically zero when p = n − 2. This imposes a condition on the right-hand side. Here the picture is more complicated than in the ˆ=0 previous subsection. Effectively, the right-hand side is a form of obstruction to solutions of B whose vanishing fixes the conformal gauge of the solution. There are three ways to approach the issue. We can expand the right-hand side of the second equation in (5.13) when p = n − 2, ˆ00 using (3.11) and (3.16), or we can we can by-pass this and instead expand the equation 0 = B ˆ = 0 given a solution of B ˆ ⊥ = 0. We choose observe that the latter is equivalent to solving trg B ˆ that it is equivalent to the latter approach because we see immediately from the definition of B (1.10), which serves here to replace the conditions A = 0 or Q = 0 which were used to fix the conformal gauge when n = 4. We expand 8(n − 2) ˆ = 4|E|2 − A2 trg B g g (n − 4) 1 = x4 |h′′x |2hx − 2 trhx h′′x · h′x · h′x + trhx h′x trhx h′′x · h′x + |h′x |2hx trhx h′′x 2 1 − trhx h′x trhx h′′x 2 3 + x −2(n − 2) trhx h′′x · h′x + 2(n − 3) trhx h′′x trhx h′x h 2 i + LWT . + x2 (n − 2)2 |h′x |2hx − n2 − 5n + 5 trhx h′

0= −

(5.14)

If we twice x-differentiate this expression and set x = 0, we obtain 2 2 (5.15) 0 = (n − 2)2 h′ (0) h0 − n2 − 5n + 5 trh0 h′ (0) ,

23

or in other words (5.16)

3 2 tf h h′ (0) 2 = n − 7n + 14n − 9 trh h′ (0) 2 . 0 0 h0 2 (n − 1)(n − 2)

Since the numerator of the coefficient on the right-hand side has no roots in the positive integers then tf h0 h′ (0) = 0 if and only if trh0 h′ (0) = 0. Henceforth, in order to keep the analysis tractable, we choose additional conditions on free data as follows (i) trh0 h′ (0) = 0, thus tf h0 h′ (0) = 0 in view of the last paragraph and so h′ (0) = 0, (ii) Scalh0 = −(n − 2) trh0 h′′ (0), (iii) Scalh0 6= 0. The first two conditions are satisfied by Poincar´e-Einstein metrics. Without the third condition, leading terms in the analysis below would vanish and we would have to compute to higher order. If we then differentiate (5.14) three times with respect to x, set x = 0, and use condition (i) above, both sides of the resulting equation vanish and we obtain no further information. Now differentiate (5.14) k ≥ 4 times and set x = 0. The highest derivative terms are products of h′ (0) with h(k−1) (0). These products vanish by condition (i) above. The next highest derivatives have the form of products of h′′ (0) with h(k−2) (0). Conditions (ii) and (iii) allow us to use these terms to determine trh0 h(k−2) (0) in terms of lower order data. We obtain 0 = − n3 − 7n2 + 14n − 9 trh0 h′′ (0) trh0 h(k−2) (0) + (k − 2) n2 − 5n + 5 trh0 h′′ (0) trh0 h(k−2) (0) + (k − 2)(k − 3) n2 − 5n + 5 trh0 h′′ (0) trh0 h(k−2) (0) (5.17) + (k − 2)(k − 3) trh0 h′′ (0) trh0 h(k−2) (0) + (k − 2)(k − 3)(n − 3) Scalh0 trh0 h(k−2) (0) + F (h0 , h′ (0), . . . , h(k−3) (0), tf h0 h(k−2) (0)) = Pn (k − 2) trh0 h′′ (0) trh0 h(k−2) (0) + F (h0 , h′ (0), . . . , h(k−3) (0), tf h0 h(k−2) (0))

for some function F , where Pn (t) is given by (5.18)

Pn (t) := (n − 2)(n − 4)t2 + (n − 3)t − n3 − 7n2 + 14n − 9

= (n − 2)(n − 4)t2 + (n − 3)t − (n − 1)(n − 2)(n − 4) + 1 ,

and in the last line of (5.17) we have used condition (ii). We can use (5.17) to solve for trh0 h(k−2) (0) in terms of lower order data and tf h0 h(k−2) (0) provided condition (iii) above holds and provided Pn (t) has no roots that are integers t ≥ 2. Lemma 5.5. Assume conditions (i–iii) above. Then for all l ≥ 1, then trh0 h(l) (0) is deter mined by lower order data and by tf h0 h(l) (0) . Unlike the n = 4 case, there is no freedom to choose the “mass aspect”.

Proof. The l = 1 case is simply condition (i) while the l = 2 case is condition (ii). For l ≥ 3, set l = k−2 in (5.17). We only need to check that Pn (l) 6= 0. Now it is easy to see that for n ≥ 5, the quadratic expression Pn (t) has one root of either sign. To see that the positive root can never be

24

AGHIL ALAEE AND ERIC WOOLGAR

√ √ an integer, observe that Pn ( √n − 2) = (n−2)2 (n−4)+(n−3) n√− 2−(n−1)(n−2)(n−4)+1 = 2 −(n − 2)(n − 4) + (n − 3) n − 2 + 1 < −(n − 2) n − 4 − n − 2 ≤ 0 for n ≥ 6, while √ √ Pn ( n − 1) = (n − 3) n − 1 + 1 > 0. Hence, when n ≥ 6, the square of the positive root t∗ lies strictly between adjacent integers n − 2 < t2∗ < n − 1, so t2∗ cannot be an integer, and so nor can t∗ . Therefore, for n ≥ 6, Pn (l) 6= 0 for any positive integer l. For n = 5 we can check explicitly that the discriminant of Pn is 136. which is not a perfect square.2 5.4. The proof of Theorem 1.5. Proof. We begin by choosing h0 such that Scalh0 6= 0. We set h′ (0) = 0 and use Corollary 5.3 2 to obtain tf h0 h′′ (0) = − (n−3) Zh0 . In accord with condition (ii) and Lemma 5.5 of the previous 1 subsection, we choose trh0 h′′ (0) = − (n−2) Scalh0 . Now that we have determined h′′ (0), we may use this as the first step in an induction. Say for some 2 ≤ k < n − 3, we have determined h(k) (0). We may use Corollary 5.3 to determine tf h0 h(k+1) (0). We then use Lemma 5.5 to determine trh0 h(k+1) (0). When k reaches n − 3, the induction halts, but we are free to choose tf h0 h(n−2) (0). Notice, however, that while the left-hand side of equation (5.2) vanishes for s ≡ k + 1 = n − 2, the right-hand side is determined by lower order terms. Therefore one must check that the right-hand side is nevertheless zero, to ensure consistency. We will deal with this momentarily. At next order, we have from Lemmata 5.3 and 5.4 that the divergence-free part of tf h0 h(n−1) (0) is also free data, while the divergence is determined by the lower-order terms already computed. As with the previous order, we again must check a consistency condition, which is that the right-hand side of (5.2) now vanishes when s ≡ k + 1 = n − 1. Once this is done, the induction resumes, and all higher order derivatives of h(k) (0), k ≥ n, are determined iteratively in terms of the free data. It remains to check that the right-hand side of equation (5.2) vanishes when s = n − 2 and when s = n − 1. We deal first with s = n − 2. With the choices h(0) = h0 , h′ (0) = 0, and 1 Scalh0 , we have established that the h(k) (0), k < n − 2, are all determined. trh0 h′′ (0) = − (n−2) But Fefferman and Graham [12, 13] have shown that, given h(0) = h0 , there is a metric obeying the Einstein equations up to order xn−2 inclusive (and unique to that order), whose hx obeys 1 h′ (0) = 0 and trh0 h′′ (0) = − (n−2) Scalh0 , agreeing with our data. Moreover, since Poincar´eˆ ˆ = 0 a metric that Einstein metrics obey B = 0, there must be a choice of data yielding from B n−2 is Einstein to order x inclusive. Since the metric we find is uniquely determined by the data n−3 up to order x inclusive, our metric and the Fefferman-Graham metric agree to order xn−3 inclusive, and there must exist a choice of data Φ = tf h0 h(n−2) (0) in our case so that the metrics agree to order xn−2 inclusive. Then for this choice of data, the relevant right-hand side function 1 F from Lemma 5.4 must vanish. But F = F (h0 , 0, − (n−2) Scalh0 ) does not depend on Φ. Finally, we deal with s = n − 1. In this case, the right-hand side of equation (5.2) must vanish for all Φ := tf h0 h(n−2) (0), not just the value given by the corresponding term in a Poincar´eEinstein metric. But simple counting shows that the total number of x-derivatives (i.e., summed over all occurrences of x-derivatives of hx ) within any single term on the right-hand side of (5.2) cannot exceed n − 1, so any h(n−2) (0) appearing in any such term must multiply h′ (0) or not multiply any x-derivative of hx at all; e.g., it could possibly multiply Scalh0 , say. But since h′ (0) = 0 by assumption, the former possibility is excluded, while the latter possibility is ruled out by parity. That is, if one expands (3.13) using (2.7) to obtain B ⊥ as a sum of terms, each 2We thank DA McNeilly for suggesting the argument used in this proof.

25

composed of x-derivatives of hx multiplying powers of x, the sum of the number of x-derivatives ˆ ⊥ . Therefore, the sum and the power of x is even. This is also true then of the terms in B mod 2 of the number of x-derivatives (acting on hx ) in each term of (5.2) when s = n − 1 should equal n − 1 mod 2, ruling out terms of the form h(n−2) (0) · f (h0 ) (where by f (h0 ) we of course mean any function of h0 , its intrinsic connection D, etc). Hence the right-hand side F of (5.2) is independent of Φ and so depends only on (h0 , h′ (0), trh0 h′′ (0)). But, if h0 yields an 1 Scalh0 ) must vanish, and does unobstructed Poincar´e-Einstein metric, then F = F (h0 , 0, − (n−2) so independently of Φ. We remark that the condition in the last line of the proof that h0 should be data for a formal series for a Poincar´e-Einstein metric (i.e., that the Fefferman-Graham ambient obstruction tensor for h0 vanishes) always holds if the bulk dimension n is even, and holds for odd n if, for example, h0 is conformally Einstein. Also, when n is even, the argument given to rule out obstructions at order s = n − 1 combined a parity argument with an appeal to the Poincar´e-Einstein case but this appeal is really just a short-cut. One can use parity alone to complete the argument when the bulk dimension n is even. If n is odd, one can similarly show that there is no order (n−2) obstruction purely by parity considerations, without appeal to the existence of a Poincar´eEinstein metric (but of course this would not work at order (n − 1) for n odd). References [1] P Albin, Poincar´e-Lovelock metrics on conformally compact manifolds, talk at BIRS workshop 18W5108 (Banff, May 2018), available as http://www.birs.ca/events/2018/5-day-workshops/18w5108/videos/watch/201805171032-Albin.html. [2] G Anastasiou and Olea, From conformal to Einstein gravity, Phys Rev D94 (2016) 086008. [3] M Anderson, L2 curvature and volume renormalization of the AHE metrics of 4-manifolds, Math Res Lett 8 (2001) 171–188. [4] L Andersson and M Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann Global Anal Geom 16 (1998) 1–27. [5] E Bahuaud, R Mazzeo, and E Woolgar, Renormalized volume and the evolution of APEs, Geom Flows 1 (2015) 136–138. [6] E Bahuaud, R Mazzeo, and E Woolgar, Ricci flow and volume renormalizability, preprint [arXiv:1607.08558]. [7] T Balehowsky and E Woolgar, The Ricci flow of asymptotically hyperbolic mass and applications, J Math Phys 53 (2012) 072501. ¨ [8] J Bergman Arleb¨ ack, Conformal Einstein spaces and Bach tensor generalizations in n dimensions, Ling¨ oping University PhD thesis (2004), unpublished. [9] PT Chru´sciel, GJ Galloway, L Nguyen, and T-T Paetz, On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds, preprint [arXiv:1801.03442]. [10] PT Chru´sciel and M Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J Math 212 (2003) 231-264 [11] Z Djadli, C Guillarmou, and M Herzlich, Op´erateurs g´eom´etriques, invariants conformes et vari´et´es asymptotiquement hyperboliques, Panoramas et synth`eses 26 (Soci´et´e math´ematique de France, 2008). ´ [12] C Fefferman and CR Graham, Conformal invariants, in Elie Cartan et les Math´ematiques d’aujourd’hui, Ast´erisque (num´ero hors s´eries, 1985), pp 95–116. [13] C Fefferman and CR Graham, The ambient metric, Annals of mathematics studies 178 (Princeton University Press, 2012). [14] AR Gover and A Waldron, Renormalized Volume, preprint [arXiv:1603.07367]. [15] CR Graham, Volume renormalization for singular Yamabe metrics, preprint [arXiv:1606.00069]. [16] CR Graham and K Hirachi.The ambient obstruction tensor and Qcurvature, in The AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA lectures in mathematics and theoretical physics 8 (2005) 59–71.

26

AGHIL ALAEE AND ERIC WOOLGAR

[17] CR Graham and JM Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv Math 87 (1991) 186–225. [18] J Maldacena, Einstein gravity from conformal gravity, preprint [arXiv:1105.5632]. [19] J Qing On the rigidity for conformally compact Einstein manifolds, Int Math Res Not 21 (2003) 1141–1153. [20] X Wang, The mass of asymptotically hyperbolic manifolds, J Differ Geom 57 (2001) 273-299. [21] E Woolgar, The rigid Horowitz-Myers conjecture JHEP 1703 (2017) 104. Center of Mathematical Sciences and Applications, Harvard University, 20 Garden Street, Cambridge MA 02138, USA E-mail address: [email protected] Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB, Canada T6G 2G1 E-mail address: [email protected]